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G = D247S3order 288 = 25·32

The semidirect product of D24 and S3 acting through Inn(D24)

metabelian, supersoluble, monomial

Aliases: D247S3, D6.2D12, C24.24D6, D12.20D6, Dic3.13D12, C8.9S32, (S3×C8)⋊2S3, C3⋊C8.25D6, C6.9(S3×D4), (S3×C24)⋊1C2, C6.9(C2×D12), (C3×D24)⋊10C2, C32(C4○D24), (S3×C6).21D4, (C4×S3).37D6, C2.14(S3×D12), C324(C4○D8), D125S33C2, C31(D83S3), C325Q167C2, D12.S36C2, (C3×C12).52C23, (C3×C24).23C22, C12.72(C22×S3), (C3×Dic3).26D4, (C3×D12).6C22, (S3×C12).45C22, C324Q8.5C22, C4.48(C2×S32), (C3×C6).36(C2×D4), (C3×C3⋊C8).32C22, SmallGroup(288,455)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D247S3
C1C3C32C3×C6C3×C12S3×C12D125S3 — D247S3
C32C3×C6C3×C12 — D247S3
C1C2C4C8

Generators and relations for D247S3
 G = < a,b,c,d | a24=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a12b, dcd=c-1 >

Subgroups: 562 in 133 conjugacy classes, 40 normal (30 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×3], C6 [×2], C6 [×4], C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C32, Dic3, Dic3 [×6], C12 [×2], C12 [×2], D6, D6 [×2], C2×C6 [×3], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×3], C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6 [×6], C4×S3, C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C4○D8, C3×Dic3, C3⋊Dic3 [×2], C3×C12, S3×C6, S3×C6 [×2], S3×C8, C24⋊C2 [×2], D24, Dic12 [×3], D4.S3 [×2], C2×C24, C3×D8, C4○D12 [×2], D42S3 [×2], C3×C3⋊C8, C3×C24, S3×Dic3 [×2], D6⋊S3 [×2], S3×C12, C3×D12 [×2], C324Q8 [×2], C4○D24, D83S3, D12.S3 [×2], S3×C24, C3×D24, C325Q16, D125S3 [×2], D247S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C22×S3 [×2], C4○D8, S32, C2×D12, S3×D4, C2×S32, C4○D24, D83S3, S3×D12, D247S3

Smallest permutation representation of D247S3
On 96 points
Generators in S96
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 36)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)(49 57 65)(50 58 66)(51 59 67)(52 60 68)(53 61 69)(54 62 70)(55 63 71)(56 64 72)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 73)(48 74)

G:=sub<Sym(96)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,73)(48,74)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,73)(48,74) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,40),(21,39),(22,38),(23,37),(24,36),(49,88),(50,87),(51,86),(52,85),(53,84),(54,83),(55,82),(56,81),(57,80),(58,79),(59,78),(60,77),(61,76),(62,75),(63,74),(64,73),(65,96),(66,95),(67,94),(68,93),(69,92),(70,91),(71,90),(72,89)], [(1,17,9),(2,18,10),(3,19,11),(4,20,12),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,33,41),(26,34,42),(27,35,43),(28,36,44),(29,37,45),(30,38,46),(31,39,47),(32,40,48),(49,57,65),(50,58,66),(51,59,67),(52,60,68),(53,61,69),(54,62,70),(55,63,71),(56,64,72),(73,89,81),(74,90,82),(75,91,83),(76,92,84),(77,93,85),(78,94,86),(79,95,87),(80,96,88)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,81),(32,82),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,73),(48,74)])

45 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E12F12G24A24B24C24D24E···24J24K24L24M24N
order122223334444466666668888121212121212122424242424···2424242424
size116121222423336362246624242266224446622224···46666

45 irreducible representations

dim111111222222222222444444
type+++++++++++++++++++-+-
imageC1C2C2C2C2C2S3S3D4D4D6D6D6D6D12D12C4○D8C4○D24S32S3×D4C2×S32D83S3S3×D12D247S3
kernelD247S3D12.S3S3×C24C3×D24C325Q16D125S3S3×C8D24C3×Dic3S3×C6C3⋊C8C24C4×S3D12Dic3D6C32C3C8C6C4C3C2C1
# reps121112111112122248111224

Matrix representation of D247S3 in GL4(𝔽73) generated by

51800
552300
0010
0001
,
485400
292500
0010
0001
,
1000
0100
00721
00720
,
436000
133000
0001
0010
G:=sub<GL(4,GF(73))| [5,55,0,0,18,23,0,0,0,0,1,0,0,0,0,1],[48,29,0,0,54,25,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,1,0],[43,13,0,0,60,30,0,0,0,0,0,1,0,0,1,0] >;

D247S3 in GAP, Magma, Sage, TeX

D_{24}\rtimes_7S_3
% in TeX

G:=Group("D24:7S3");
// GroupNames label

G:=SmallGroup(288,455);
// by ID

G=gap.SmallGroup(288,455);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,135,142,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^24=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^12*b,d*c*d=c^-1>;
// generators/relations

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×
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